Ozone groups and centers of two-parameter quantum Heisenberg algebras

Let $p,q$ be roots of unity with $pq\neq1$, and put $\ell=|\langle p,q\rangle|$ and $t=\ord(pq)$. Up to isomorphism, the two-parameter quantum Heisenberg algebras $\mathcal H_{p,q}$ are exactly the noetherian PI graded down-up algebras, so everything here holds for that whole class. The ozone group